Problem 63 - Algebra 2

Question

Divide. State any restrictions on the variables. \(\frac{7 a x^{3}}{8 b y^{2}} \div \frac{14 a x^{4}}{4 b y}\)

Step-by-Step Solution

Answer

The quotient is \(\frac{1}{4x y}\) with restrictions \(b \neq 0, y \neq 0\) and \(x \neq 0\).

Step 1: Convert Division Into Multiplication

Rewrite the expression \(\frac{7 a x^{3}}{8 b y^{2}} \div \frac{14 a x^{4}}{4 b y}\) as \(\frac{7 a x^{3}}{8 b y^{2}} \cdot \frac{4 b y}{14 a x^{4}}\) where the division operation has been converted to multiplication by the reciprocal of the second fraction.

Step 2: Simplify the Multiplication

Now, multiply the numerators together and the denominators together. \(\frac{7 a x^{3}}{8 b y^{2}} \cdot \frac{4 b y}{14 a x^{4}}\) becomes \(\frac{7 a x^{3} \cdot 4 b y}{8 b y^{2} \cdot 14 a x^{4}}\), which simplifies to \(\frac{28 a b x^{3} y}{112 a b x^{4} y^{2}}\).

Step 3: Simplify the Resulting Fraction

Simplify the fraction \(\frac{28 a b x^{3} y}{112 a b x^{4} y^{2}}\) by cancelling out common terms from the numerator and denominator. This gives \(\frac{1}{4x y}\).

Step 4: State the Restriction on Variables

The restriction on the variables arises from the requirement that the denominators of the original fractions and the final answer cannot be zero. Thus, the restrictions are \(b \neq 0, y \neq 0\) and \(x \neq 0\).

Key Concepts

Restrictions on Variables

When dealing with rational expressions, it's crucial to remember that we cannot divide by zero. This fundamental rule affects which values our variables can take. If a variable makes the denominator zero, that variable value must be excluded from valid solutions. For the expression \(\frac{7 a x^{3}}{8 b y^{2}} \div \frac{14 a x^{4}}{4 b y}\), we need to ensure that the denominators in both fractions and any intermediate steps are not zero.

Therefore, we set restrictions to avoid division by zero:

• Since \(b\) is a factor in the denominators, \(b eq 0\).
• As \(y\) and \(y^2\) appear, \(y eq 0\).
• The presence of \(x\) in the denominator of the simplified answer requires \(x eq 0\).

Observing these restrictions ensures that our expressions remain valid mathematical statements.

Fraction Division

Dividing fractions might seem a bit tricky, but once you grasp the flip-and-multiply method, it becomes much simpler. In essence, dividing by a fraction is the same as multiplying by its reciprocal. For the problem \(\frac{7 a x^{3}}{8 b y^{2}} \div \frac{14 a x^{4}}{4 b y}\), we rewrite it as a multiplication problem:

- The original division is transformed into \(\frac{7 a x^{3}}{8 b y^{2}} \cdot \frac{4 b y}{14 a x^{4}}\), where we've flipped the second fraction \(\frac{14 a x^{4}}{4 b y}\) to get its reciprocal \(\frac{4 b y}{14 a x^{4}}\).This transformation is the key step in fraction division, making it easier to subsequently multiply the fractions and then simplify the result.

Simplification of Algebraic Expressions

Simplification of algebraic expressions means reducing them to their simplest form while maintaining their value. It's essential because it makes expressions easier to work with and understand. In our exercise, after rewriting the division as multiplication, we have the expression: \[\frac{7 a x^{3} \cdot 4 b y}{8 b y^{2} \cdot 14 a x^{4}}\] Here's how you simplify:

• Multiply the numerators, and then multiply the denominators.
• Cancel out common factors from the numerator and denominator. The term \(a x^3\) in the numerator and denominator, as well as \(b\) in both parts, can be reduced.

After removing the common terms, we further simplify it to \(\frac{1}{4xy}\), ensuring the fraction is as concise as possible. The process of simplification not only assists in solving the problem but also solidifies understanding by peeling away layers to reveal the core relationship of the algebraic expression.